In many circuits it is desirable to provide a temperature-compensated reference current from which other currents can be derived. The classic current mirror circuit with a "V.sub.BE " circuit for temperature compensation is shown in FIG. 1a. A regulated voltage V.sub.reg is applied at terminal 10 to the emitters of transistors 15 and 20. Transistor 15 is configured as a diode with its base tied to its collector, and its collector current is denoted as the reference current I.sub.ref. The collector current of transistor 20 is denoted as I.sub.1, which is supplied to a load (passive or active). Provided transistors 15 and 20 are matched, there will be a prescribed relationship between I.sub.ref and I.sub.1. For example, if the emitter areas of transistors 15 and 20 are equal, then ideally we have I.sub.1 =I.sub.ref.
A "V.sub.BE " circuit 25 and resistor 30 connect the collector of transistor 15 to ground. V.sub.BE circuit 25 may simply be one or more diodes providing a voltage drop of NV.sub.BE, where N is an integer and V.sub.BE is the forward voltage drop of a diode. For simplicity, we will assume hereinafter that all diode forward voltage drops and transistor base-emitter voltage drops are equal to V.sub.BE. Also, for simplicity of discussion, we will assume that the emitter areas of transistors 15 and 20 are equal to each other. Generalization of the following discussion for unequal emitter areas is obvious.
Ignoring the base currents of transistors 15 and 20, the reference current I.sub.ref will equal the current through circuit 25 and resistor 30, which we denote as the trim current I.sub.trim. Then, the reference current is given by the simple relationship L.sub.ref =(V.sub.reg -(N+1)V.sub.BE)R, where R is the resistance of resistor 30. Thus, for a given N and V.sub.reg, the resistor 30 sets the reference current I.sub.ref, as well as I.sub.1.
By proper choice of N, some degree of temperature compensation in the circuit of FIG. 1a can be achieved. For example, assume that V.sub.reg has no appreciable variation due to temperature, and that the temperature dependencies of V.sub.BE and R are known. Furthermore, assume that for a temperature T near a nominal temperature T.sub.0, R is given by the linear function R=R.sub.0 (1+k(T-T.sub.0)) where R.sub.0 is the resistance of R at temperature T=T.sub.0 and k is a thermal coefficient of resistance.
The derivative of I.sub.ref with respect to temperature T for the circuit of FIG. 1a for the above assumptions is given by dI.sub.ref /dT=-(dV.sub.BE /dt)(N+1)/R-kR.sub.0 (V.sub.ref -(N+l)V.sub.BE )/R.sup.2. Setting dI.sub.ref /dT=0 in this expression at the nominal temperature T.sub.0, and denoting the value of V.sub.BE at temperature T.sub.0 as V.sub.0, we obtain the following expression for N for which the circuit of FIG. 1a is properly temperature compensated: N=V.sub.reg /(V.sub.0 -(1/k)dV.sub.BE /dt)-1.
However, this expression will in general not be satisfied for N being an integer. One approach to achieve temperature compensation for a circuit of the type shown in FIG. 1a is simply to choose N=where denotes the closest integer to x. A more accurate solution would be to use the "V.sub.BE multiplier" circuit 25' as shown in FIG. 1b in place of circuit 25, in which case the voltage drop from the collector of transistor 15 to node 35 would be V.sub.BE (1+R.sub.1 /R.sub.2), where R.sub.1 and R.sub.2 are the resistances of resistors 45 and 40, respectively. With the V.sub.BE multiplier circuit, the resistances of R.sub.1 and R.sub.2 are chosen so that (1+R.sub.1 /R.sub.2)=V.sub.reg /(V.sub.0 -(1/k)dV.sub.BE /dt)-1.
However, even with the use of a V.sub.BE multiplier circuit, the above analysis for the circuit of FIG. 1a or 1b is still somewhat idealized. In practice, because transistors 15 and will not be exactly matched, emitter resistors would be used with transistors 15 and 20. Furthermore, often it is desirable to provide a number of mirrored currents, in which case a number of transistors would have their bases connected to the collector of transistor 15. However, with a number of transistors connected to transistor 15, the effect of the total base current cannot be ignored, and ;ef would no longer be sufficiently close to I.sub.trim. In this case, it is well known in the art to use a "beta-helper" transistor, or buffer, to reduce the effect of the base current upon the relationship between I.sub.ref and I.sub.trim. Thus, a more practical circuit for current mirroring used in the prior art is shown in FIG. 2, where, for example, we have included three transistors 20, 21, and 22 for providing the three currents I.sub.1, I.sub.2, and I.sub.3, the emitter resistors 51, 52, 53, and 54, and the "beta-helper" transistor 50.
For the circuit of FIG. 2, if the base current of transistor 50 is ignored, then the reference current I.sub.ref is given by I.sub.ref =(V.sub.reg -V.sub.BE (3+R.sub.1 /R.sub.2))/(R+R.sub.e), where R.sub.e is the resistance of emitter resistor 51. For temperature compensation, the values of R.sub.1 and R.sub.2 can be chosen so that the derivative of I.sub.ref with respect to temperature is close to zero. However, this simplified analysis of the circuit in FIG. 2 is only approximately true, and the circuit of FIG. 2 suffers from variation in the mirrored currents I.sub.1, I.sub.2, and I.sub.3 due to temperature. It should be noted that the previous expression for I.sub.ref is only approximately true because the base current of transistor 50 has been ignored. Furthermore, the base current of transistor 50 increases as more transistors such as transistors 20-22 are used to supply additional mirrored currents. The base current of transistor 50 could be ignored provided its beta value is high.
In addition to the problem of a small beta value of transistor 50, a potentially more serious problem with the prior art circuit of FIG. 2 is that the total voltage drop from V.sub.reg terminal 10 to node 35 must be 3V.sub.BE or greater. This problem will limit the operation of the circuit of FIG. 2 to a regulated voltage source with a voltage at least as large as RI.sub.trim +3V.sub.BE, and it may not be possible to temperature compensate the reference current I.sub.ref. This latter issue is best illustrated by writing the previous equation for I.sub.ref as I.sub.ref =(V.sub.reg -xV.sub.BE )/r, where x=3+R.sub.1 /R.sub.2 and for simplicity we have lumped R and R.sub.e into r so that r=R+R.sub.e. Taking the derivative of I.sub.ref with respect to temperature T(to first order the derivative of R.sub.1 /R.sub.2 with respect to temperature is zero provided R.sub.1 and R.sub.2 have equal temperature coefficients) and setting the result equal to zero yields: ##EQU1## However, depending upon the temperature thermal coefficient of the effective resistance r and the regulated voltage V.sub.reg, the solution to the above equation may be less than three, i.e., x&lt;3, which cannot be realized by the circuit of FIG. 2 because x=3+R.sub.1 /R.sub.2 &gt;3. Thus, the circuit of FIG. 2 can provide a temperature compensated reference current only if ##EQU2##
Another temperature compensation scheme is illustrated by the circuit of FIG. 3. A regulated voltage V.sub.reg is applied at terminal 60 and a current is supplied to load 62. The current supplied to load 62 is set by the regulated voltage and by resistor 64. Provided the V.sub.BE 's of transistors 66 and 68 are sufficiently matched, the current flowing through resistor 64 will be V.sub.reg /R.sub.64, where R.sub.64 is the resistance of resistor 64. The current flowing through resistor 64 is mirrored by transistors 70 and 72. However, although the temperature variations in the base-emitter voltage drops of transistors 66 and 68 are canceled, resistor 64 will have a temperature variation, and therefore the current supplied to load 62 may not be sufficiently temperature compensated.